Two Answers That Both Work on the Number-Analogy Warm-Up
A Grade 4 teacher writes three number analogies for a ten-minute warm-up and projects each with four options. Twenty-four students answer on whiteboards.
Item 1: 3 → 12 as 5 → ? Options 14, 15, 20, 60. Key: 20. Item 2: 4 → 7 as 10 → ? Options 3, 13, 17, 40. Key: 13. Item 3: 5 → 15 as 2 → ? Options 6, 10, 12, 45. Key: 6.
Results: on Item 1, nine students choose 20, eleven choose 14, four choose 15. On Item 2, twenty-one choose 13. On Item 3, thirteen choose 12, eight choose 6, three choose 10.
A colleague reviews the results and says the students who chose 14 and 12 do not understand analogies and need reteaching on 'apply the same rule both times.' The teacher is not convinced the items deserve that conclusion and wants to check the items before planning any reteach.
1. For each of the three items, decide whether it has one defensible answer or two. Make the competing rules explicit.
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Apply the rule 'same operation, applied once, in both pairs' to every option.
Item 1. 3 → 12 is +9 or ×4. Both survive: 5 + 9 = 14 is an option and 5 × 4 = 20 is an option. Two defensible answers, so the item is double-keyed. 15 and 60 fit no uniform rule (60 = 5 × 12, a 'multiply by the first pair's second term' error).
Item 2. 4 → 7 is +3 (or ×1.75). 10 + 3 = 13 is an option; 10 × 1.75 = 17.5 is not. The other options (3, 17, 40) fit no uniform rule. One defensible answer, so the item is sound.
Item 3. 5 → 15 is +10 or ×3. 2 + 10 = 12 is an option and 2 × 3 = 6 is the key. Two defensible answers, so the item is double-keyed.
Two of the three items, not the students, are the problem. The eleven students who chose 14 and the thirteen who chose 12 applied the taught principle correctly.
2. Rewrite Item 1's option set so that exactly one option is defensible, and state the principle behind your choice.
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Replace 14 and keep three others: 15, 20, 25, 60. Check each against at least two candidate rules before printing.
- 20 = 5 × 4, the intended ×4 rule.
- 15 = 5 + 10 or 5 × 3; neither pairs with a rule that takes 3 to 12 (3 + 9 = 12, but 5 + 9 = 14; 3 × 4 = 12, but 5 × 3 = 15). Safe.
- 25 = 5²; attracts the 'square the second number' error, since 3 is not 12². Safe and diagnostic.
- 60 = 5 × 12; attracts 'multiply by the first pair's second term'. Safe.
Principle: for any multiplicative analogy a → ka, the additive partner is always constructible, because a′ + (b − a) always works. An option set is valid only if it excludes that value. This cannot be fixed by choosing 'better' numbers, only by checking the options. Write down both the additive and the multiplicative rule for every item before it is copied.
3. Write the reply you would give the colleague, and say what you would tell the eleven students.
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'The same rule both times' is exactly what the eleven students did: 3 + 9 = 12 and 5 + 9 = 14 is one operation applied twice; 3 × 4 = 12 and 5 × 4 = 20 is another. The principle does not choose between them. Those students did not misunderstand analogies; they were defeated by an option set containing two correct answers. The same holds for 12 on Item 3.
What I would tell the students: check every option before committing; if two options work, say so, and prefer the one built on the relationship between the numbers themselves (here multiplication) rather than on a constant difference. Then say plainly that this is a convention of multiple-choice tests, not a rule of arithmetic.
And I would not reteach. Reteaching a correct answer teaches children to guess the teacher's intent instead of the mathematics.